The sling accelerator.

This puzzle is much like "the skater problem" found in some physics textbooks. Contrary to naive assumptions, when the string wraps around the stationary axle, the mass on the end of the string does not increase its speed. This can be seen in two equivalent ways:

The string wraps around the axle without being moved along the instantaneous length of its straight portion. Therefore the axle does no work on the string, and the string cannot do work on the ball.
The string is always perpendicular to the path of the ball. Therefore its tension has no component in the direction of motion, and cannot change the ball's tangential velocity.

Therefore the speeds (and kinetic energies) of the ball at points A and B are the same. Release of the axle catch exerts no torque on the ball, so the ball begins the second cycle with the same speed (and kinetic energy) with which it began the first cycle. That's the ideal case that assumes no energy loss to dissipative processes. There's no energy gain per cycle.

In this kind of spiral motion the angular speed or the ball about the center of the axle does increase, while the ball's moment of inertia about the center of the axle decreases.

Follow up question: Discuss the ball's angular momentum changes per orbit.

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